Ore On Functions and Craphs Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the y-axis, the origin, or neither. \( h(x)=3 x^{2}-x^{10} \) Determine whether the function is even, odd, or neither. Choose the correct answer below. odd \( 10,1.3 .41 \) neither even

To determine if the function \( h(x) = 3x^{2} - x^{10} \) is even, odd, or neither, we evaluate \( h(-x) \). Calculating, we find: \( h(-x) = 3(-x)^{2} - (-x)^{10} = 3x^{2} - x^{10} = h(x) \). Since \( h(-x) = h(x) \), the function is even. The symmetry in the graph of \( h(x) \) confirms this, as it's symmetric with respect to the y-axis. For a little fun, think of even functions as having a mirror-image buddy on the other side of the y-axis, while odd functions do a wild pirouette around the origin! In this case, our function is rock-solid in its evenness. By recognizing patterns in polynomial functions, you can quickly determine symmetry—if the powers of x in your function are all even, congrats, it’s even! If they're all odd, then it’s odd. Mixing them up? That’s neither!